1. Faster, Higher, Stronger Alexander Mitov Theory Division, CERN * * A(lso)-K(nown)-A(s): “Citius, Altius, Fortius"

2. Talk Surely Inspired by Olympic principles, but mostly by hadron collider challenges Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 Collider Phenomenology: where are we and where do we go from here? Top physics: demanding and challenging The role of soft gluon resummation What can we hope to achieve? Towards NNLO. Dimuons, tt-bar FB, etc.

3. The nature of the problem Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 What we do is try to quantify the equation Experiment – SM = ? In practice, we phrase it like Experiment – LO = ? Or Experiment – NLO = ? Does it make a difference; does it matter how we phrase our questions? We do not pay too much attention because we all believe we know what we mean; But do we, really? It is not a gamble to predict that these “nuances” will be central to the LHC program 3 examples 

4. Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 Drell-Yan: vector boson rapidity distribution Notice how meaningless a representation of the total uncertainty the scale variations is ! Collider Phenomenology: where are we and where do we go from here? Anastasiou, Dixon, Melnikov, Petriello ‘03

5. Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 Differential Higgs production Notice how meaningless a representation of the total uncertainty the scale variations is ! Collider Phenomenology: where are we and where do we go from here? Anastasiou, Melnikov, Petriello `04

6. Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 F_B asymmetry in tT+jet Collider Phenomenology: where are we and where do we go from here? Dittmaier, Uwer, Weinzierl ‘07 Scale variation at LO suggests uncertainty well below 10% Scale variation at NLO grows substantially NLO corrections around – 100% Scale variation completely misrepresents the theoretical uncertainty!

7. Collider Phenomenology: where are we and where do we go from here? Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 This discussion might not appear very relevant if it was not for the fact that it is at the center-stage, today. Again. F-B asymmetry in tT production W+2jets anomaly: “hey, what is happening in 2jets” Same-sign dimuon excess at the Tevatron (D0)

8. The NLO revolution Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 Finally, we have reached the point of “everything at NLO”. (very impressive recent talk by Frixione @ CERN) It is a truly great development. But why did it take so long? All essential ingredients are known for at least 15 years. A Phys. Rep. type of overview appeared 3 days ago (a great reading!): Ellis, Kunszt, Melnikov, Zanderighi Here is an excerpt from the Abstract: The success of the experimental program at the Tevatron re-inforced the idea that precision physics at hadron colliders is desirable and, indeed, possible. The Tevatron data strongly suggests that one-loop computations in QCD describe hard scattering well. Extrapolating this observation to the LHC, we conclude that knowledge of many short-distance processes at next-to-leading order may be required to describe the physics of hard scattering … Do we still need to justify using NLO ??

9. On the wings of the NLO revolution and beyond Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 My point is: LO, or NLO or NNLO, etc. is irrelevant. What matters is the level of control we have over theory. So, I argue, going to NNLO in many case is a necessity, that can also be a source of deep satisfaction. And a disclaimer: What I do not do, and am not advocating, is to go for partial and unquantifiable higher order corrections. What I do is to calculate the next largest source of uncertainty. The line between the two is fine, and I will be sure to point it out in examples.

10. Top-pair production at hadron colliders Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

11. Top – some history Top-pair production is known within NLO/NLL QCD Main feature: Very large NLO corrections ~50% Appears in many bSM models Great prospects for experimental improvements down to 5% How to do NNLL resummation is now completely understood Is NNLL resummation/threshold approximation a substitute for the NNLO? Threshold approximation to NNLO also available Czakon, Mitov, Sterman ’09 Beneke, Falgari, Schwinn ‘09 Beneke, Czakon, Falgari, Mitov, Schwinn ‘09

12. Is the threshold region dominant? The observed cross-section is an integral over the product of:  Partonic cross-section,  Partonic flux. Very large NLO corrections! Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 Pure LO vs. pure NLO

13. Is the threshold region dominant? Important for the partonic cross-section (as expected) Not for the flux. Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 Czakon, Mitov ‘09

14. Top-pair cross-section: 2-loop threshold expansion Beneke, Czakon, Falgari, Mitov, Schwinn ‘09 Derive NNLO threshold approximation for the cross-section Use soft-gluon expansion (from resummation) Extract 2-loop Coulombic terms (from, say, e+e-  tT) Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 Czakon, Mitov, Sterman ’09 Beneke, Falgari, Schwinn ‘09

15. Top-pair total X-section: Tevatron numbers Being an approximation, how robust are these numbers? Try to understand the physics; Stress-test in all possible ways; Quantify the sensitivities. Construct a number of NLO+NNLL and NNLO_aprox “scenarios” to analyze: Plotted for each scenario are: central values scale uncertainty Used independent variation of: renormalization scale factorization scale See Cacciari et al ‘08 Defined on the next slide NNLL Resummation Approximate NNLO Best prediction For m_top=171GeV Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

16. Top-pair total X-section: Tevatron numbers NNLL resummation Approximate NNLO Best prediction Two approaches to NNLO_approx (depends on how the unknown const. are treated) ① Unknown constants AND log(mu) terms are omitted: Larger scale variation Consistent approximation Uncertainty = scale variation ② Unknown constants’ log(mu) terms INCLUDED: Much smaller scale variation (~ the true NNLO) Uncertainty = scale variation AND constant variation Constant varied in a “reasonable range” Both approaches are mutually consistent Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

17. Top-pair total X-section: Tevatron numbers NNLO approximations vs. exact NLO vs NLO approximations NLO/NNLO Pdf sets consistently used Reduction in sensitivities going from NLO to NNLO Exact NLO NLO: threshold approximation LO+NLL NNLL/NNLO_approx approximations For m_top=171GeV Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

18. With: Czakon; Beneke et al; Cacciari, Mangano, Nason; Moch, Uwer Top-pair total X-section: Tevatron numbers Preliminary data Published data Prediction based on NLO+NNLL / NNLO_approx Scale and PDF variation MSTW2008 (NNLO) Note O(10%) decrease compared to MRST2006 (NNLO) Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 Reconcile past differences and come up with common theory systematics

19. Top-pair total X-section: LHC @ 14 TeV numbers NLO/NNLO Pdf sets consistently used Great reduction in sensitivities expected at full NNLO Various NLO/NLL approximations NNLL/NNLO_approx approximations For m_top=171GeV Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 LHC 7 – very similar!

20. Threshold approach: the conclusions Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 Once we get to NNLO, resummation is not as important (tower of soft logs is ~0 beyond NNNLO) A consistent use shows reduction in theoretical uncertainty Recall: theory uncertainty = scale variation + constant variation This has been neglected in the past – and bold conclusions were made It also shows the potential for improving theory once NNLO is known: up to factor of 2 (Tevatron) – down to ~ 3% up to a factor of 3 (LHC) – down to ~ 2-3% First pointed out by Bonciani, Catani, Mangano, Nason`98

21. Towards NNLO for tT production and more Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 Work in progress with M. Czakon

22. Towards NNLO Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 There are 3 principle contributions: 2-loop virtual corrections (V-V) 1-loop virtual with one extra parton (R-V) 2 extra emitted partons at tree level (R-R) And 2 secondary contributions: Collinear subtraction for the initial state One-loop squared amplitudes Korner, Merebashvili, Rogal `07 Known, in principle. To be done numerically.

23. Towards NNLO: V-V Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 Required are the two loop amplitudes: qq  QQ and gg  QQ. Their high energy limits and their poles are known analytically The qq  QQ amplitude is known numerically Numerical work underway for the gg  QQ Czakon, Mitov, Moch ’07 Czakon, Mitov, Sterman ‘09 Ferroglia, Neubert, Pecjak, Yang ’09 Czakon `07 Czakon, Bärnreuther

24. Comments about the 2-loop amplitudes Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 Czakon’s calculation is in principle straightforward (highly tedious): Derive a large set of masters, Derive differential equations for them (2-dim pde) Derive numerically boundary conditions in the high energy limit (Mellin-Barnes) Solve numerically the pde’s That works for qq  QQ and for gg  QQ This will work for any 2-to-2 amplitude, obviously. But hardly beyond. What’s the future here? I believe that right now this is the biggest (and perhaps only) obstacle for NNLO phenomenology on a mass scale. Two-loop unitary – is it a mirage?

25. All is done The method is general. Explicit contribution to the total cross-section given Towards NNLO: R-R Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 Czakon `10-11

26. Towards NNLO: R-V Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 The idea here is to get the NLO revolution to work All that is needed are the finite terms of the amplitude. No subleading terms in Epsilon needed (use counterterms - more later). Existing packages can handle any problem (in principle). Speed seems to be an issue (even with OPP) An analytical evaluation in terms of masters might be needed. The poles of any 1-loop amplitude can be written analytically. So, in principle, all is known. The only missing piece are the subtraction terms in the soft and collinear limits

27. Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 Why subtractions? We need to integrate over the real gluon. That generates additional divergences when the gluon is soft and/or collinear to external legs. Idea: devise a CT which approximates |amplitude|^2 in the soft/collinear limits: |M|^2 = |M|^2 – CT + CT Finite in all limits. Can be integrated numerically in 4d. Note: the poles of the amplitude are known. Assumed subtracted beforehand. Singular phase-space integration Simple function Towards NNLO: R-V

28. Counter-terms for 1-loop amplitudes Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 Collinear limits are easy: emissions off massive lines are finite; massless - known Soft limit is an open problem. We have solved it; paper to appear. How to devise CT for any 1-loop amplitude (masses and all )? Bierenbaum, Czakon, Mitov - to appear. Bern, Del Duca, Kilgore, Schmidt ‘98-99 Soft limit of 1-loop massive amplitudes Consider an (n+1) - point amplitude, with one external gluon (momentum q) When the gluon becomes soft (q  0) the amplitude becomes singular: J : the soft-gluon (eikonal) current. It is process-independent!

29. The one-loop soft-gluon current Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 So, the singular (soft) limit of 1-loop amplitudes is controlled by J: Bern, Del Duca, Kilgore, Schmidt ‘98-99 Catani, Grazzini `00 The one-loop correction J^(1) is known in the massless case: Process independent calculation of eikonal diagrams:

30. The one-loop soft-gluon current Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 Result expressed in terms of 3 scalar integrals (simple, complicated, harsh) Note: these are not scaleless integrals! M_1 and M_2 through Gauss hypergeometric functions. M_3 is very hard. Involves multiple polylogs (also appear in one-loop squared) Calculate 3 kinematical configurations: (i,j) massive/massless, or (i,j) incoming/outgoing Understood the analytical continuation spacelike  timelike. Number of checks (small mass limit; poles; numerical checks)

31. The one-loop soft-gluon current Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 Here is the result for the IM parts of the one-mass case (simplest thing to show): The leading q-dependence factors out in d-dimensions; the rest is homogeneous!

32. Derivation of CT Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 Then, the square of the Born diagram becomes: … and the one-loop amplitude reads: The current reads:where: Note!

33. Comment on subtractions (dipole or otherwise) Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011 Dipoles are not enough beyond NLO (known from many places now) FKS: no shift in variables; no divergent integration. Works similarly to sector decomposition. FKS does not really care about the color structure (which is very involved); It is all about the kinematics. So, CT are derived. What’s next? Integrate them over phase space. It is a problem. I first tried Catani-Seymour type of approach. Realized fast that Nature cannot be that cruel. The way out seems to be FKS. No details at that point, just few comments:

34. Summary and Conclusions  We have developed methods for NNLO calculations in any 2-to-2 process.  Very soon to produce results for the tT-total cross-section at Tevatron.  That will be followed by F-B asymmetry and fully exclusive observables.  Applications for dijets and heavy flavor production in DIS at NNLO.  Methods are fully exclusive and numeric.  Down the road – produce partonic Monte Carlo at NNLO. For the tT total cross-section:  Discussed relation between fixed order calculations and resummation in tT  Applicability of threshold approximation (many speculative statements out there)  Potential for tT at NNLO is at the 2-3% level. Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

35. Backup Slides Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

36. Singularities of Massive Gauge Theory Amplitudes Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

37. Amplitudes: the basics  Gauge theory amplitudes: UV renormalized, S-matrix elements  The amplitudes are not observables:  UV renormalized gauge amplitudes are not finite due to IR singularities.  Assume they are regulated dimensionally d=4-2  Explicit expression for the IR poles of any one-loop amplitude derived Catani, Dittmaier, Trocsanyi ‘00 The small mass limit is proportional to the massless amplitude Note: predicts not just the poles but the finite parts too (for m  0)! Mitov, Moch ‘06 Becher, Melnikov ‘07 What was known before (massive case): Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

38. Factorization: “divide and conquer“ Structure of amplitudes becomes transparent thanks to factorization th. Note: applicable to both massive and massless cases I,J – color indexes. J(…) – “jet” function. Absorbs all the collinear enhancement. S(…) – “soft” function. All soft non-collinear contributions. H(…) – “hard” function. Insensitive to IR. Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

39. For an amplitude with n-external legs, J(…) is of the form: Factorization: the Jet function i.e. we associate a jet factor to each external leg. Some obvious properties: - Color singlets, - Process independent; i.e. do not depend on the hard scale Q. J i not unique (only up to sub-leading soft terms). A natural scheme: J i = square root of the space-like QCD formfactor. Sterman and Tejeda-Yeomans ‘02 Scheme works in both the massless and the massive cases. The massive form-factor’s exponentiation known through 2 loops Mitov, Moch ‘06 Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

40. Factorization: the Soft function Soft function is the most non-trivial element (recall: it contains only soft poles). But we know that the soft limit is reproduced by the eikonal approximation.  Extract S(…) from the eikonalized amplitude: The LO amplitude M(…) The eikonal version of the amplitude. (the blob is replaced by an effective n- point vertex) Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

41. Factorization: the Soft function Calculation of the eikonal amplitude: consider all soft exchanges between the external (hard) partons LO 1-loop 2-loops The fixed order expansion of the soft function takes the form: … as follows from the usual RG equation:  All information about S(…) is contained in the anomal’s dimension matrix  IJ Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

42. the Soft function at 1 loop Here is the result for the anomalous dim. matrix at one loop The massless case O(m) corrections in the massive case where: - all masses are taken equal, - written for space-like kinematics (everything is real). Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

43. The Soft function at 2 loops The simplest approach is the following. Start with the Ansatz: Reproduces the massless case Parametrizes the O(m) corrections to the massless case Then note: the function P (2) ij depends on (i,j) only through s ij  P (2) ij = P (2) (s ij ) This single function can be extracted from the known n=2 amplitude: the massive two-loop QCD formfactor. Bernreuther, Bonciani, Gehrmann, Heinesch, Leineweber, Mastrolia, Remiddi `04 Gluza, Mitov, Moch, Riemann ‘09 Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

44. The Soft function at 2 loops Kidonakis ’09 Becher, Neubert ‘09 Czakon, Mitov, Sterman ‘09 The complete result for the 2E reads: This term breaks the simple relation from the massless case! Above result derived by 3 different groups: Kidonakis derived the massive eikonal formfactor; Becher, Neubert used old results of Korchemsky, Radushkin Aybot, Dixon, Sterman ’06 Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

45. The Soft function at 2 loops What about the 3E contributions in the massive case? Until recently there existed no indication if they were non-zero! In particular, the following squared two-loop amplitudes are insensitive to it: Known numerically Czakon ’07 Poles reported Czakon, Bärnreuther ’09 3E correlators not vanish if at least two legs are massive – direct position-space calculation for Euclidean momenta (numerical results) Mitov, Sterman, Sung ’09 Exact result computed analytically Ferroglia, Neubert, Pecjak, Yang ’09 Czakon, Mitov, Sterman ‘09 Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

46. The Soft function at 2 loops. Massive case. The types of contributing diagrams: The analytical result is very simple: where: Ferroglia, Neubert, Pecjak, Yang ’09 A.M., Sterman, Sung ‘10 The calculation of the double exchange diagrams is very transparent. Agrees in both momentum and position spaces Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

47. Massive gauge amplitudes: Summary  The results I presented can be used to predict the poles of any massive 2-loop amplitude with:  n external colored particles (plus arbitrary number of colorless ones),  arbitrary values of the masses (usefull for SUSY).  Results checked in the 2-loop amplitudes:  Needed in jet subtractions with massive particles at 2-loops  Input for NNLL resummation (next slides) Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

48. The connection to resummation at hadron colliders Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

49. How is the threshold resummation done?  Only soft emissions possible due to phase space suppression (hence kinematics)  That’s all there is for almost all “standard” processes: Higgs, Drell-Yan, DIS, e + e - The resummation of soft gluons is driven mostly by kinematics: Sterman ‘87 Catani, Trentadue ‘89 In top pair production (hadron colliders) new feature arises: Color correlations due to soft exchanges (n>=4) Key: the number of hard colored partons < 4 Non-trivial color algebra in this case. Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

50. The top cross-section: NNLL resummation Factorization of the partonic cross-section close to threshold: N – the usual Mellin dual to the kinematical variable that defines the threshold kinematics: J’s – jet functions (different from the ones in amplitudes) S,H – Soft/Hard functions. Also different. Drell-Yan t-tbar total X-section t-tbar – pair invariant mass Kidonakis, Sterman ‘97 Czakon, Mitov, Sterman ‘09 Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

51. The top cross-section: NNLL resummation Specifically, for top-pair production we have: where:  – is the Drell-Yan/Higgs cross-section  – observable dependent function (i.e. depends on the final state) Defines the poles of the massive QCD formfactor in the small-mass limit. Bernreuther, Bonciani, Gehrmann, Heinesch, Leineweber, Mastrolia, Remiddi `04 Gluza, Mitov, Moch, Riemann ’09 Mitov, Moch ‘06 Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

52. The top cross-section: NNLL resummation Note: the Soft function satisfies RGE with the same anomalous dimension matrix as the Soft function of the underlying amplitude! Here is the result for the Soft function: Therefore: knowing the singularities of an amplitude, allows resummation of soft logs in observables! Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

53. The top cross-section: NNLL resummation We also need to specify a boundary condition for the soft function: For two-loop resummation we need it only at one loop (since its contribution at two loops is only through the running coupling). For example, for the total t-tbar cross-section in gg-reaction it reads: Can be derived by calculating the one-loop eikonal cross-section. Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

54. The top cross-section: NNLL resummation Combining everything we get the following result for the resummed total t-tbar cross-section: Hard function. Known exactly at 1 loop. And the anomalous dimension is: Czakon, Mitov, Sterman ’09 Beneke, Falgari, Schwinn ‘09 Jet functions (from Drell-Yan/Higgs) Czakon, Mitov ’08 Hagiwara, Sumino, Yokoya ‘08 Fixed by the small-mass limit of the massive formfactor! Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011

55. Get the cross-section How we put all this to work?  Match fixed order and resummed results:  RESUM =  NLO +  SUDAKOV -  OVERLAP   NLO   SUDAKOV is known exactly, : anomalous dimensions and matching coefficients needed. Now known at NNLO Known at NLO i.e. at present one can derive the NLO+NNLL cross-section Faster, higher, stronger Alexander Mitov CERN, 27 June, 2011